Question

The perpendicular distance from the origin to the focal chord drawn through the point \( (4, 5) \) to the parabola \( y^2 - 4y - 3x + 7 = 0 \) is:

• A. \( \frac{2}{5} \)
• B. \( \frac{1}{\sqrt{2}} \)
• C. \( \frac{1}{5} \)
• D. \( 1 \)

Hint:

When dealing with parabolas and focal chords, use the properties of the parabola’s equation and the geometry of the chord to calculate distances.

Answer 1

The Correct Option is C

The given parabola is \( y^2 - 4y - 3x + 7 = 0 \). First, complete the square to rewrite the equation in a standard form: \[ y^2 - 4y = 3x - 7 \] Complete the square for \( y \): \[ (y - 2)^2 = 3(x - \frac{7}{3}) \] This is a standard parabola equation with vertex \( (\frac{7}{3}, 2) \). Step 1: The point \( (4, 5) \) lies on the parabola. Using the properties of the focal chord, the equation for the distance from the origin to the focal chord is derived. Step 2: After performing the calculation, the perpendicular distance from the origin to the focal chord is \( \frac{1}{5} \). % Final Answer The perpendicular distance is \( \frac{1}{5} \).